Mathematical Sciences: Applications of Nonstandard Analysis to Measure Theory, Potential Theory, and Related Topics

数学科学：非标准分析在测度论、势论及相关主题中的应用

• 批准号: 8702064
• 负责人: Peter Loeb
• 金额: $ 4.45万
• 依托单位: University of Illinois at Urbana-Champaign
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 1987
• 资助国家: 美国
• 起止时间: 1987-07-01 至 1989-12-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=8702064&HistoricalAwards=false
• 关键词: Mathematical; Sciences; Applications; Nonstandard; Analysis

Professor Loeb is one of the world's leading experts in the applications of the fundamental foundational concept in mathematics called nonstandard analysis. This construct, first introduced by logicians, is an ingenious rigorous development of the classical concepts of infinitesimally large and small numbers. Nonstandard analysis, therefore, is a mathematical vindication of Leibnitz' philosophical viewpoint in his invention of the calculus. Professor Loeb is the leading practitioner in the use of nonstandard methods to prove "standard" results in a broad range of mathematical analysis and applications. Loeb's work in nonstandard measure theory and nonstandard measure theory and nonstandard probability theory has led to important contributions to probability theory, mathematical physics, and mathematical economics. The notion of Loeb measure, introduced by the principal investigator, has become a standard tool in probability theory, stochastic analysis, and potential theory. The current proposal by Professor Loeb has two parts. First, he proposes to combine his nonstandard integration theory with nonstandard convex hulls. Applications are envisaged to Banach spaces and Banach space valued stochastic processes. Secondly, he proposes to continue his work on the concept of ideal (Loeb) boundary, developed for very general potential theories. This research promises to have important implications for classical potential theory.

勒布教授是世界领先的数学基本概念非标准分析应用专家之一。这个概念最早是由逻辑学家提出的，它巧妙而严谨地发展了经典的无穷大和无穷小数的概念。因此，非标准分析是对莱布尼茨在微积分发明中的哲学观点的数学证明。勒布教授是使用非标准方法在广泛的数学分析和应用中证明“标准”结果的领先实践者。勒布在非标准测度论、非标准测度论和非标准概率论方面的工作对概率论、数学物理和数学经济学做出了重要贡献。Loeb测度的概念由主要研究者提出，已成为概率论、随机分析和位势理论中的标准工具。勒布教授目前的提议包括两个部分。首先，他提出将他的非标准积分理论与非标准凸壳相结合。应用于Banach空间和Banach空间值随机过程。其次，他建议继续他关于理想(Loeb)边界概念的工作，该概念是为非常一般的势能理论而发展的。这项研究有望对经典势论产生重要的影响。